Hoeffding's inequalities: A counterexample

A sequence : ₀ satisfiesHoeffding's inequality of order n if wheneverX₁,...,X_n are independent nonnegative integer-valued elementary random variables and are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX₁,...,X_n. We show that for each integerm greater than 2 there exists a sequence satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.